Introduction to Fortranfluid.web.nitech.ac.jp/Gotoh_Home_page/Edu/Under...line : 1 line can be...

Nagoya Institute of Technology


Graduate School of Engineering

Department of Scientific and Engineering Simulation

T. Gotoh

Introduction to Fortran


Fortran (FORmula TRANslator)

Advanced programming language for numerical computation

developed by IBM (1950)

・ Oriented for mathematical operation

(matrix algebra, complex number, various Built-in functions)

・ Better performance than C, C++ for the numerical computation

・ Many mathematical library

・ Easiness for parallelization

・ High performance on super computer


History

Fortran66 Born as the first programming language

Fortran77 Fixed form, many mathematical libaries developed

(vector type super computer)

Fortran90 Major change, free form, structuring

enhanced parallel operation

(parallel type super computer)

Fortran95 Minor change

Fortran2003 Refinement (including Fortran90/95）

Fortran2008 Standardization of parallel operation

(massive scalar parallel machine)


Simple example (free form)

! Computation of three kinds of averages comment

Program Example_1_1 program sentence (not necessary)

Real:: a,b,av1,av2,av3 declaration of type

(all variables need type declaration)

print *, ‘enter two numbers’ output to the standard device (*: display)

Read *, a,b Read data （*: Keyboard)

av1=(a+b)/2.d0 computation

av2=(a*b)**0.5

av3=2.0/(1.0/a+1.0/b)

print *, av1,av2,av3 write data on the display

End Program Example_1_1 End


Process

1. Write program by using an editor application

Save it as test.f90 (.f90, fortran90 free format)

2. Compile program

Use command prompt

Type g95 test.f90

３. Execute

Type as a.exe

Fortran getting started


Change directory

Compile

execute


Fortran grammar

1


Character

alphabet : A-Z (non case sensitive, keep easiness of reading )

number : 0～9

underline : _

special ： (space) ＝＋－＊ / ( ) ，． ’ ：！ ” ％＆；＜＞？$

Writing

fixed form： begin with 7th column and end up to 72

file name is sample.for (Windows) sample.f （Unix)

free form： begin at any position

file name is sample.f90


Free form

line : 1 line can be started at any column and should be shorter than 132 character

; semicolon allows to add executable statements after this in the same line

x=1.d0 ; y=2.d0 ; z=3.d0

comment : part of line after ! becomes comment (even in the middle of line)

! This is an example

y=sin(x) ! Sine function

continuation : to continue to the next line, add & at the end of the previous line,

y= 1.0+ 2.0*x*x+ cos(x) &

+ 0.5*sin (2.0*x)

space: space can be inserted exept executable statements

○ endif ○ end if

○ forall × for all


Program main

…..

call time_advance (u1,u2,u3)

…..

End program main

Subroutine time_advance (u1,u2,u3)

……

return

End

main program

sub program

Program unit Example

Program unit

Program structure

Program <program name> (can be omitted)

…….

…….

end <Program program name>

Kind of sub program name of subprogram

(necessary)

…….

…….

end <Kind of sub program name of subprogram >


Kind of statements

declaration statetment : set memory size, non executable statement

real*8 :: a,b,c ; integer :: i,j,k

dimension (10,10):: array

executable statement : control computation and flow of processes

do i= 1,10

a(i)=2*i

if (i > 3) exit

end do

Order of statements

program unit name （program, subroutine, function）

declaration (type statement, common statement)

data

executable statement

contains

end (need for each program unit)


Name of program unit and variables begin with alphabets and less than 31 letters

vel1, vel2, stream_function, Potential_1

random_number_generator

Variables and constants variables ： quantity whose value chanegs as program proceeds

scalar, vector, matrix

integer, real single, double real, complex …

constants： quantity uncanged under processes integer constat, 4 bytes (8×4=32 bit) -231=-2147483648 ～ 231-1= 2147483647 real consant, single precision 4 bytes, double precision 8 bytes decimal form 3.141592 floating form 7.199847e-12 complex constant, set of two real numbers 2+3i=(2.e0, 3.e0) logical constant, 4 bytes .true. .false. character constant, 1 character (alphabets) 1 byte 1 character (Japanese) 2 bytes


Loop label

loop1 : do j=1,n2

loop2 : do i=1,n1

…………..

…………..

end do: loop2

end do: loop1

Effective when the operation part is long


Substitution

variable = variable, constant, formula

= to substitute the value of the right hand side

Program summation

integer :: sum,i

sum=0

do i=1,100

sum = sum+1 algorithm for summation

end do

print *, sum

end

Numerical computation


Arithmetic operation

additon + a + b

subtraction – a - b

multiplication * a * b

division / a / b

power ** x**3 = x*x*x

Order of arithmetic operation

addition, subtraction < product, division < exponent

compute from left to right. When ( ) exits, () must be computed first.

x+y*z= x+(y*z)

x/y/z= (x/y)/z

x*y**2= x*(y**2)

x**y**2=x**(y**2)


Output of characters

print *, ` what you write‘

real, parameter :: pi=3.14159265

real:: r,s

read *,r

s=pi*r*r

print *, `radius, area`, r, s

end

Oder of operation

From top to bottom

integer :: s

s=10

s=s+1; print *, s

s=s+1; print *, s

s=s+1; print *, s

end


Exercise １

read positive real x and positive integer n, compute x**n and x**(1.e0/n),

and print them

Exercise 2

Find Fahreheit from Celcius

F=1.8C +32.0

Exercise 3

Compute BMI from weight L (m) and W (kg).

BMI = w / L2


! ex1

real:: x,xn,xn1

integer:: n

read *,x,n

x1=x**n

xn1=x**(1.e0/n)

print *,x,n,x1,xn1

end

! ex2

real::c,f

read *,c

f=1.8*c +32.0

print *,c,f

end

! ex3

real::L,w,bmi

print *,'Enter W and L'

read *,w,L

bmi=w/(L**2)

print *,'weight[kg]=',w,', Hight[m]=',L, ', BMI=',bmi

end

Answer Program


2


Control of flow

if, stop, goto, case

1 If

if ( Iogical statement, relation, logical algebra) excutable statement

when the value of ( ) is ture, the statetment is executed.

Integer n, s

n=32

s=1

if ( n>=10 ) s=2*s +1

2 stop stop <‘message’>

stop here (with message) without jump to the end statement

Integer n

if ( n<0 ) stop ‘n must be positive integer’


3. goto

goto number (go to number)

jump to the line with specifiled numer

Integer:: n, s, k, sum

k=0; s=0; sum=0

10 print *,’enter students mark (quit, if negative integer)‘

read *, n

if ( n<0 ) go to 999

s=s+1

sum=sum+n

if ( n >=60) k=k+1

go to 10

999 print *,‘Total and No . of students passed this exam. =‘,s,k

print *,’Average of score =‘, sum/s

end


Relational operator

> .gt. greater than

< .lt. smaller than

<= .le. smaller than or equal

== .eq. equal to

/= .ne. not equal to

if ( n+m > 0 ), if ( a < 4.5 )

types of valiables in both side in relational operator should be the same

Logical operator

.not. logical negation

.and. logical product

.or. logocal sum

.eqv. logical equaivalence

.neqv. logical nonequivalence

n=0 or 1 (n==0).or.(n==1)

0 < k < 8 (0<k).and.(k<8)

x=y=z (x==y).and.(y==z)


Priority of operators

.neqv. .eqv. < .or. < .and. < .not.

< relational operator < arithmetic operator

use of ( ) is recommended

operations of the same priority should be made from the left

x+y > 1.0e0 (x+y) > 1.e0

( x < 3.14 ) .or. ( z < sqrt(2.e0) )


if statement with more funcions label1: if ( logical statement ) then

………

executable statements

………

endif <labbel1> (use the label in pair)

if (logical statement ) then

………

else

………

endif

if (logical statement_1 ) then

………

else if ( logical statement_2 ) then

………

else

………

endif

1. no limitation on the number of branches

2. multiple if nests can be used

3. no enter into if nest is allowed

No enter exit allowed


block1: if (logical statement_1 ) then

………


………

end if blcok2

………

end if block1


………


………

end if blcok1

………

end if block2

Nest structure of if statements


! Calculating various functions

real :: x

print *,'input a real number:'

read *,x

if(x>0.0) then

print *, sqrt(x), log(x), log10(x)

else

print *,' Not defined for x<0‘

end if

end

Intrinsic function

Example 1


0 2 1

1

! Triangle pulse real:: x,f print *,'input x:' read *,x if(x<0.0) then f=0.0 else if (x<1.0) then f=x else if (x<2.0) then f=-x+2.0 else f=0.0 end if print *, x,f end

x<0 is already excluded



Example 2


case nest

for many divided case <loop1:> select case ( formula ) case ( value or range of formula ) ！A executable statements case (value or range of formula ) ！B executable statements case default ！（otherwise） executable statements end select <loop1>

No overlap in divided case


integer :: i print *, ‘ Enter mark ‘ read *, I if ( i<0 .or. i>100 ) stop select case (i) case (90: ) print *, ‘S’ case (80:89) print *, ‘A’ case (70:79) print *, ‘B’ case (60:69) print *, ‘C’ case default print *, ‘D’ end select end

Example


Intrinsic function

Fundamental functions used in mathematical operatios

ｘ, y：real、ｚ：complex、 i, j, k ： integer

general name：return value according to type of argument x

ｘ：single real exp(x)：single real

ｘ：double real exp(x)：doube real

ｚ：single complex exp (z) ：single complex

ｚ：double complex exp (z) ：double complex

exp (x)

general name argument

sqrt(x)

abs(x)

sin(x)

exp(x)

log(x)

log10(x)

z=cmplx(x,y)

real(z)

aimag(z)

conjg(z)

aint(x) xの整数部を実数で返す

int(x) xの整数部を整数で返す

anint(x) xを四捨五入して実数で返

す

Nint(x) xを四捨五入して整数で返

す


Intrinsic function

aint(x) return integer part of x as real number

int(x) return integer part of x as integer number

anint(x) return real number of rounded x

Nint(x) return integer number of rounded x


! Calculating various functions real :: x,y complex::z print *,'input a real number:' read *,x if(x>0.0) then print *, sqrt(x), log(x), log10(x) else print *,' Not defined for x<0‘ end if print *,'input a complex number x,y (=x+iy):‘ read *, x,y z=cmplx(x,y) print *, sqrt(z), exp(z), log(z) end

Example


Note definition of complex functions of complex argument

Example for z= i


Example for x＝2 , (x,y)=(0,1)


Exercise 4

Read positive real number and print the integer part and decimal part, separately.

Exercise 5

Enter real balues of a, b, and c for the quadratic equation ax2+bx+c=0,

and find solutions in real number according to the discriminant.


! ex4

real:: x,y

integer:: n

read *,x

n=int(x)

y=x-aint(x)

print *,x,n,y

end

! ex5

real::a,b,c,d,x1,x2

print *,'Enter a,b,c'

read *, a,b,c

d=b*b-4.0*a*c

if(d>=0) then

x1=0.5*(-b+sqrt(d))/a

x2=0.5*(-b-sqrt(d))/a

print *, x1,x2

else

print *, 'No real solution'

endif

end

Answer


3


Variables need declaration of type

integer :: variable list (single precision integer 4 bytes) real :: variable list (single precision real 4 bytes) real*8 :: variable list (double precision real 8 bytes) complex :: variable list (single precision complex 8 bytes) complex*16 :: variable list (double precision complex 16 bytes)

real :: x,y

real*8 ::d1,d2

complex::z

integer :: n,m,l

Data type


Set initial values

(once to be done when program unit is called)

Real :: pi=3.14159265,s,r

Integer :: n=30

Print * ‘Enter r’

Read *, r

S=pi*r*r

n=n+1

Print *, n,r,s

End

Real :: pi=3.14159265,s,r

Integer :: n

Print * ‘Enter r’

Read *, r

S=pi*r*r

n=30

n=n+1

Print *, n,r,s

End

Left : 30 is set for n only one time

right : 30 is always subsituted in n

give initial values in the type statement


change types

x=real(k)

n=integer(s)

Z=cmplx(a,b)

Use intrinsic function

Implicit type declaration

Variables with the first character starting with

i,j,k,l,m,n automatically interpreted as integer

Kmin, Kmax, jnum, num

Others real


!----- Solar year -----

real ::year,day,jikan,hun,byo

integer :: d,h,m

year=365.2422

day=aint(year)

d=int(day)

jikan=24.0*(year-day)

h=int(jikan)

hun=60.0*(jikan-aint(jikan));

m=int(hun)

byo=60.0*(hun-aint(hun))

print*,'1yearis',d,'days',h,'hours',m,'minutes', &

byo,'seconds‘

end


Nullify of implicit type

types of all the variables need to be declared

effective to find bugs.

Change of implicit type

implicit real*8 (a-h, o-z) double precision for variables beginning with

a-h, o-z

implicit complex (z) complex for variables beginning with z

Implicit type (firts character [ range of characters]), .....

implicit none


Logical type declaration

logical :: flag, index, range

flag=.true.

index=.false.

range =(8.0<x).and.(x<10.0)

Character type an logical type

character type declaration needs type and length(the number of characters)

character (len=8) :: c=‘doraemon’ c: character variable, `doraemon’ is content

character (2) :: fname, file, a

character ([len=]No,) :: variable list

character :: variable1*length1, variable2*length2, .....

logical :: variable list


Program Logical_example

Implicit none

logical :: hantei

real:: a=0.e0, b=1.e0, x

read *, x

hantei=(a<x).and.(x<=b)

print *, hantei When it is true, value of decision is `T`

End false, value of decision is `F`

Example


Named constants

parameter declaration to name constants used in program

integer, parameter :: mask=2147483467

real , parameter :: pi=3.14159265, pi2=2.e0*pi

character(1), parameter :: sp=‘ ‘, ap=“’”

character(len=*), parameter :: f=‘This is Fortran95’

* set the length to the actual length used

Type, parameter :: constant＝value [or formula]


Complex, high precision

complex :: variable list

complex*16 :: variable list

real*8 :: variable list

real([kind=]8) :: variable list

real(8) :: deg,rad,s,c

real(8),parameter :: pi=3.141592653589793

print*,'Input angle variable in degree:‘

read*, deg

rad=pi*deg/180.0

s=sin(rad)

c=cos(rad)

print*,'SIN=',s,', COS=',c

print*,'COS^2+SIN^2=',s**2+c**2

end


!----- Complex number ----- real :: x complex :: z integer :: n complex, parameter :: i=(0.0,1.0) ! print*,'Input a real number x and an integer n:' read *,x,n; z=exp(i*x) print*,z**n print*,cos(n*x),sin(n*x) end


Mixed operation and substitution

・for shorter variables, 0s are appended to the lower digits

・to substitute a real number for complex variable, imaginary part is set to be 0

change to the stronger type and then operate and substitute

integer < real < complex, single < double < quadruple

integer/integer returns integer

7/3 2

1/3 0

2**(-2) 1/(2**2) 0

N-m*(n/m) surplus when n is divided by m

X**n x*x*....*x

X**y exp(y*log(x))


Substitution of different types

real :: a,b real*8 :: x a=0.0 b=2.14 x= a+b a+b addition in 4 bytes

0s are appended to lower digits to be 8 bytes

print *, a,b,x end

to compute right hand side first, to change the type of result to that of

the right hand side, and then to substitute

Real ::x

X=1.0e0/3 1.e0/3.e0 x

X=(3.14e0, 2.56e0) only real part is substituted for x

X=5 substitute 5.e0 for x


Real::x

Real(8) ::y

y=1.0e0/3.e0 1.e0/3.e0

compute in single precision and append 0s to lower digits and substitute for y

y=2.0d0/3.e0 2.0d0/3.0d0

compute in double precision and append 0s to lower digits and substitute for y

integer ::i

i=-3.623 -3 is substituted for i

i=(3.14e0, 2.56e0) 3 for i

i=5.0e0/3 1 for i

complex ::z

z=1.0e0 z=(1.0e0, 0.0e0)

z=5.e0 z=(5.0e0, 0.0e0)


Type of function and argument

general function name is used

y=f(x) : type of f is determined according to type of x

and change the results of f to that of y

x: double precision f: double precision

x: double precision complex f: double precision complex


Exercise 6

1. Change 0.1e0 to double precision and compare the result with 0.1d0

2. For an integer n, print its square root computed in single and double precision.

Compute the square of the square root of a single precision real number,

and change it to that in double precision. What is difference between the original

and results?

3. Read two real numbers x and y, and make a complex number z=x+ i y.

Compute z^2 and 1/z and print them.


4


Iteration

powerful ability of computers (no tired)

do loop

［label：］ do do-variable=initial, limit [, increment]

・・・・

end do ［label］

integer :: n,sum=0

do n =1,100

sum=sum+n*n

end do

print *,sum

end

defalt １

negative value is also allowed

do n =1,100,2

do n=2,100,2

do n=100,1, -1

do n=1,2, -1 When the initial value exceeds the limit value in the driection of increment, no execution

After exiting the loop, n=101


loop1: do j=1,n

………

loop2: do i=1,m

………

end do loop2

………

end do loop1

Multiple do loop and nesting

loop1: do j=1,n

………

loop2: do i=1,m

………

end do loop1

………

end do loop2


Exiting loop

do i=1,m

………

if( a> eps) exit

………

end do

………

exit

cycle

do i=1,m

………

if( a> eps) cycle

………

end do

Jump to the line after the end do line

Jump to the End do line


integer :: m,n,mn

var1:do m =2,9

var2: do n =1,9

print *,m,'*',n,'?'

read *,mn

if(mn==m*n)then

print *,‘Correct'

else

print *,‘wrong'

end if

end do var2

end do var1

end


Format for input and output

For easiness of reading large amount data

Print format, output list

Write(*, format) output list

* ： standard (input : key board, output : display)

‘(list of format item)’ (list of format item) is interpreted as character constant

character :: fmt*21

..........

fmt=‘(_1x,_i3,_2x,_3F10.5)’ 21 characters including (, space , )

print fmt, k,s,c,p

.........

number format (list of format item) any place within a program unit is allowed

print 100, k,s,c,p

100 format ( 1x, i3, 2x, 3F10.5)

Format


Format field descriptor integer : w, d, n, r

Iw for integer data, w : width (in character unit), stuck from the right I5 : 56789

Fw.d for real data, w : width, stuck from the right,

d : digits after decimal poin w≥ d+2

F5.2 ： □2.35 , -4.12

Ew.d for real data in the floating form with exponent of 10 w≥ d+7

Dw.d for real double precision data in the floating form with exponent of 10 w≥ d+7

Aw for character data w : width, stuck from the right A5 : abcde

A adjust to the actual number of characters

r ….. Iterate by r times (….)

nX skip by n spaces. can not omiteven when n=1 1x : □ 3x : □ □ □

‘……’ write directly string in the format ‘This is an example.’

“……” the same as the above ‘(1x, I3,1x, “Hx=“, F5.3)’

d

w E12.5: -0.23452E-01

5

12


continued

/ change line

( ) repeat ‘(/, 3(1x, e12.5))’

print *

write(*,*) write a blank line

read * for characters ‘AKB48’

read ‘(A)’ just type as you want Hello


!ex5-3 !----- Table of SIN and COS ----- real, parameter :: rad=3.1415926/180.0 real :: r,s,c,p integer ::k print *,"kakudo SIN COS CHECK“ do k=0,90,10 r=k*rad s=sin(r) c=cos(r) p=s*s+c*c print '(1x,i3,2x,3f10.5)',k,s,c,p end do end


!ex5-4 !----- Mean and Standard Deviation ---- integer :: n=0,mark real :: w=0.0,v=0.0,mean,sd do print *,'score ( 0<x end)' read *,mark if (mark<0)exit n=n+1 w=w+mark v=v+mark**2 end do if(n>0)then mean =w/n; sd=sqrt(v/n-mean**2) print '("average=",f5.1)',mean print '("sd=",f5.1)',sd end if print '("total=",i5)',n end

mean

standard deviation


Avoid Infinite loop

!ex5-5 !---- the greatest common divisor --- integer :: m,n,k print *,'Imput two integer m,n(>0):' read *,m,n if(m>0 .and. n>0)then do k=mod(m,n) if (k==0) exit m=n n=k end do if (n/=1)then print *,‘the greatest common divisor =',n else print *,'nothing' end if end if end

Guarantee to exit from do loop


3からの奇数をqまで虱潰しに調べる．このqは以下のようにして求め

ることができる．いま， qまで調べたとすると、このqの整数倍は素数

ではない．すなわち q*1,q*2,q*3,…, q*m は除外される．こうし

て調べられるmの最大はqまで．mがq以上ならmとqを取り換えればよい．

このq*qがn以上であればすべて網羅したことになるので、 q*q ≥ nと

見積もれる．

n

1

2

3

1 2 3 m

q 3 n

! --- Prime Number upto 100 --- Integer :: n, m, q, max=1000 print *, 2 ! no search for even numbers except 2

loop: do n = 3, max, 2 q=Nint(sqrt(real(n))) do m = 3, q, 2 ! Examine all the odd numbers up to q

if( mod(n,m)==0 ) cycle loop ! When it is divided, it is not prime number

! and exit the loop

end do print *, n ! When the loop is passed, it is a prime number

end do loop ! Jump to here by cycle statement

end

Suppose the prime numbers up to q are found.

Since mulitples of q like q*1, 2*2, q*3 are not prime numbers,

Those numbers are excluded from the search for numbers >q.

The maximum number m examined in this way is q.

When n is greater than q, switch m and q.

When q*q >=n, all the possible numbers are examined

and thus search is finished


do while

[label] : do while ( logical formula)

…………

end do [label]

do i=1,100

………

end do

i=1 do while (i<=100) ……… i=i+1 end do

i=1 do while (i<=100) i=i+1 ……… end do

The case i=101 is executed Caution! the position of i=i+1

equivalent


!ex 5.7 ! Computation of sqrt(x) by recursive equation implicit none Real:: r=0.e0, x, d=1.e-5 Do print *,'Input a positive real number' read *, r if(r>0.e0) exit End do x=r do while (abs(x**2-r)/r >d) x=0.5e0*(x+r/x) end do print '(1x,e12.5)’, x end

Convergence criterion：


Exercise 7

The Fibonacci sequence is given by

Find x_n for n=1, ...,50 and print n and x_n in one line

Compute the ratio x_n/x_{n-1} and observe whether the ratio converges as n increases.


Simple formula

Trapezoidal rule

Simpson’s rule

Numerical integration

x0 x1 xj xn

h

f j

f(x)

x


1. Calculate analytically (leave algebra on note)

2. Numerically integrate by two methods and compare the results with

the anaytical results obtained in 1.

3. Observe the convergence by changing the number of grid points

Simple rule and torapozidal rule

Torapezoidal rule and Simpson‘s rule

Torapezoidal rule and Simpson‘s rule

when the upper limit is infinite, choose a finite value L and increase it gradually

and see the convergence. It is also useful to plot the function and to see how fast

the function decays.

Exercise 8


5


Array

Vector, Matrix

Declaration of array （1 dim.）

type, dimension([bottom: ] top) :: list of array names

type:: array1([bottom: ] top) , array2([bottom: ]top) , ….

Integer, dimension (1:100) :: number,mask

Real :: x, r, a(-100:100)

Complex, dimension(-100:0) :: zk, wk

Character, (len=1) :: fname(1:20)

default １

x and r are scalars

size＝top – bottom + 1

number 101

a 201

zk 101

fname 20

array element : specify by the number

33rd element of number : number(33)

-4th element of zk : zk(-4)


Initial setting of array

Constant array (/const, const, …/)

Do type iteration ( / (data list, do variable=initial, limit[, increment] ) / )

Integer :: a(0:9)=(/0,1,2,3,4,5,6,7,8,9/)

Integer :: n, a(0:9)=(/(n,n=0,9)/), b(1:10)=(/(2**k,k=1,10)/)

Real, dimension(0:2) :: x=(/1.0e0, 2.0e0,. 3.0e0/)

Integer ::n

real ::w(0:2)=(/(0.5e0**n, n=0,2)/)

Real::x, s(0:7)

x=2.0e0

S=(/(x,n=1,8)/)

Array structure specification (/data list/)

(/(x,n=1,8)/) can be used in the

execution statement


Operations for array

Simultaneous operations: substitution, addition, subtraction, multiplication

(1) Partial array

Array name ([initial index] : [limit] [:increment )

Integer :: a(1:5)=(/2,4,6,8,10/)

a(4:) a(4:5) (/8,10/)

a( :3) a(1:3) (/2,4,6/)

a( : ) a(1:5) (/2,4,6,8,10/)

a(1:5:2) (/2,6,10/)

a(5:1:-1) (/10,8,6,4,2/)

a(3:3) (/6/) a(3:3) 1 dim. array with 1 element

Real :: x, a(5)=(/(2**n, n=1,5)/)

a(3:3)=x OK

x=a(3:3) NO x: scalar, a: 1dim. Array


(2) Array substitution

Array＝other array with the same shape ( Shape: size and dimensionarity)

Integer :: n, a(1:10), b(1:5), c(1:16)

a=(/(n,n=0,9)/)

b=a(1:5)

c(1:10)=a

c(12:16)=a(6:10)

c(11)=1

c(11:11)=1

b=0 all elements are ０

c(1:10:2)=1 odd numbers elements are１

c(2:10:2)=0 even number elements are ０


i

1

2

1 2

j

i+1 n

i

n

Find maximum

swap

!----- Sorting Array Elements -----integer::m,mark(100),n=0,max,maxi,i,j do while(n<=99) print*,"Input integer,or negative one to stop:" read*,m; if(m<=0) exit n=n+1; mark(n)=m end do !----- Sorting do i=1,n-1 max=mark(i) maxi=i do j=i+1,n if(mark(j)>max) then max=mark(j) maxi=j end if end do if(maxi/=i) then mark(maxi)=mark(i) mark(i)=max end if end do !----- Output do i=1,n print"(' Rank',i3,' =',i10)",i,mark(i) end do end

Sorting


(3) Array arithmetic operations

Short expression of the operations for elements in terms of array names alone

Integer, dimension(1:50) :: sum, dif, pro, div &

even=(/(2*i, i=0,49)/), odd=(/(2*i+1, i=0,49)/)

Real, dimension(1:20) :: a, b=(/(2.0e0**n,n=0,19)/)

sum=even+odd sum( : )=even( : )+odd( : )

dif=even-odd dif( : )=even( : )+odd( : )

pro=even*odd pro( : )=even( : )*odd( : )

div=even/odd div( : )=even( : )/odd( : )

a=sqrt(b) a( : )=sqrt( b( : ) )

real::x(0:4)=(/(2.0**n,n=0,4)/)

x=x(0:4)*x(4:0:-1)

print*,x

end


(4) Vecrtor operation

Simultaneous operations for the arrays

Integer :: a(0:9)=(/0,1,2,3,4,5,6,7,8,9/)

Reults of A differ from those of B ！

a(1:9)=a(0:8)

do i=1,9

a(i)=a(i-1)

end do

a(0) a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9)

a(0) a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9)

0 0 1 2 3 4 5 6 7 8

a(0) a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9)

time

time

0 0 0 0 0 0 0 0 0 0

A

B


(5) Array inputs and outputs

to use array name or to use do statetment

Integer :: a(10)=(/(n,n=1,10)/)

print '(10I5)', (a(i), i=1,10)

print '(10I5)', a

print '( 5I5)', a(1:5)

print '(10I5)', (a(i), i=10,1,-1)

print *

do i=1,10

print '(10I5)', a(i)

end do

end

10 elements in one line

10 lines each of which contains one element


!----- Pascal's Triangle -----

integer, parameter::max=12

integer::i, c(0:max)=0

c(1)=1

do i=1,max

c(1:i)=c(1:i)+c(0:i-1)

print'(i2,":",50i5)',i,c(1:i)

end do

end program

１

１１

0 １２１ 0

１３３１

１４６４１

i

max


Multi dimensional array

type, dimension ([bottom1:] top1, [bottom2:] top2, ....) :: array list

type:: array1 ([bottom1:] top1, [bottom2:] top2, ....) , ….

real :: a(1:3,1:3), x(1:3), y(1:3), z(1:3)

x=a(1,1:3) or x=a(1,:)

y=a(2,1:3) or y=a(2,:)

z=a(3,1:3) or z=a(3,:)

Order of elements and structure on the memory of multidimensional array

real :: a(1:3,1:2)

メモリー上に記録される仕方

a(1,1) a(1,2)

a(2,1) a(2,2)

a(3,1) a(3,2)

do i=1,10000

do j=1,10000

a(i,j)=i+j

end do

end do

do j=1,10000

do i=1,10000

a(i,j)=i+j

end do

end do

The same results but the left is faster than the right


Dynamic allocation of arrays

type, allocatable :: array1(:, :, ….) list

type, dimension (:, :, …) , allocatable :: array list

……

allocate (array1（range of size）, …. )

……

deallocate(array1, …. )

integer :: upper_limit

integer, allocatable :: freq(:)

.....

read*, upper_limit

allocate(freq(1:upper_limit))

.....

deallocate(freq)

Allocate the required arrays in execution and deallocate after completed

real, dimension(:,:), allocatable :: x,y

.....

n=1000

allocate(x(n,n),y(2*n,2*n))

.....

deallocate(x,y)


!----- Matrix Product ----- integer,dimension(:,:),allocatable::a,b,c integer::m=0,n=0,k=0,i,j,h ! do while(m<=0.or.n<=0.or.k<=0) print*,"Input Array Size m, n, k ( > 0) :" read*,m,n,k end do allocate(a(m,n),b(n,k),c(m,k)) ! print*,"Input Elements of A in a Matrix Form:" read*,((a(i,j),j=1,n),i=1,m) print*,"Input Elements of B in a Matrix Form:" read*,((b(i,j),j=1,k),i=1,n) ! do i=1,m do j=1,k c(i,j)=0 do h=1,n c(i,j)=c(i,j)+a(i,h)*b(h,j) end do end do end do do i=1,m print"(10i5)",(c(i,j),j=1,k) end do end


where

where （selection rule）array substitution

Integer a(50),a1(50)

where ( a>=40) a1=nint(2.0e0*(a+50)/3.0e0) simultaneous substitution

Execute the substitution to the array only for the elements at which a condition is satisfied

where

［label：］ where （selection rule 1）

array substitutions

elsewhere (selection rule 2）

array substututions

end where ［label］


forall

forall （range of index，[selection rule]）array substitution in index

index＝[initial] :[limit] [:increment]

forall (i=1:1000, mark(i)>=60) mem(i)=‘pass’

forall (i=1:100, j=1:100) a(i,j)=i*j

forall (i=0:90) s(i)=sin(0.0174533e0*i)

forall (i=0:7, j=1:7, a(i)<0) b(i,j)=0

forall (i=1:9) a(i)=a(i-1) simultaneous substitution


forall(i=1 : mem, mark(i)>=60)

mark(i)=nint(2.0e0*(mark(i)-40)/3.0e0+60

list(i)=‘pass’

End forall

forall

［label：］ forall （formula for index，[selection rule]）

substitution operations with index

end forall ［label］


integer::m=0,n=0

real,allocatable::a(:,:)

do while(m<=0.or.n<=0)

print*,"Input Array Size m, n ( > 0) :"

read*,m,n

end do

allocate(a(1:m,1:n))

print*,"Input Elements of A:"

read*,(a(i,1:n),i=1,m)

where(a<0) a=0.0

do i=1,m

print"(10f8.5)",sqrt( a(i,1:n) )

end do

end


６


Character data

Interface between man and machine

Declaration of character

character (len=8) :: a,b,c=‘doraemon’

character :: fname(1:10)*8

character ([len=] length) :: variable name list

character :: name1*length, name2*length2，・・・・

character :: array([bottom:] top) *length，・・・・

Partial string of character

Part of character string

Variable name of character（[bttom] : [top]）

character (len=8) :: c=‘doraemon’ C(3:3) r r C(4:8) emon


Substitution

Length of the left hand side＝ Length of the right hand side

character (len=8) :: c

＜

D o r a e m o n

＞ D i r a c

Pading from the left side

Extra part is cut off

character (len=8) :: a=‘abcdefgh’ a(1:5)=a(2:6) ‘bcdeffgh’

a(2:6)=a(1:5) ‘aabcdegh’

K o l m o g o r o v


!---- Hexadecimal Representation ---- Implicit none Integer :: dec, i character(len=1) :: h(0:15)=(/'0','1','2','3','4','5','6','7' & ,'8','9','A','B','C','D','E','F'/) character(len=8) :: hex ext: do hex=" " print*, "input a positive integer,or 0 to stop:" read*, dec; if(dec<=0) exit ext hxd: do i=0,7 hex(8-i:8-i)=h(mod(dec,16)) dec=dec/16 if(dec==0) exit hxd end do hxd print ‘(1x,A8)’, hex end do ext end

Fill from the left

Go to the next digit

Hexadecimal expression

４０Ｅ

232=168=(24)8


Character (1) :: f(9)=(/'1','2','3','4','5','6','7','8','9'/) Character (22):: fmt Integer :: n,k n=2; k=3 Read *, int fmt="( 2x, 'Int = ', "//f(n)//"I"//f(k)//" )" Print fmt, int end

Concatination of character strings

string1 // string2

character:: a*9,f*7='Fortran', g*2='95' a=f//g Print '(1x,a9)', a f='boy' g='s' a=f//g print'(1x,a9)', a end

Fortran95

boy____s_

Print ( 2x, ‘Int = ’, nIk ), int


!---- rearrange characters of a word---- character(len=5) :: w0,w1,w2,w(1:24) integer :: i,j,k,n=1 print*,“input a word in 4 letters: " read '(a)',w0 do i=1,4 w1=w0(i:i)//w0(1:i-1)//w0(i+1:5) do j=2,4 w2=w1(1:1)//w1(j:j)//w1(2:j-1)//w1(j+1:5) do k=3,4 w(n)=w2(1:2)//w2(k:k)//w2(3:k-1)//w2(k+1:5) n=n+1 end do end do end do print '(/12a5))', w end

Sorting of characters

b l u e

u l b e

1番目に来る文字を先頭から順に選ぶ

２番目に来る文字を2番目以降から順に選ぶ


Input and output of data

On the cosole (Key board and display)

print format, output list

read format, output list

1. To store and extraction of small size of data

f=‘( A10, F10.5 )’ write(6,f) x write(*,‘( A10, F10.5 )’ ) x print‘( A10, F10.5 )’, x

f=‘( 3I10 )’ read(5,f) i,j,k read(*,‘( 3i10 )’) i,j,k read(*,100) i,j,k 100 format(3i10)

same


Input and output of data

Assignment of file name of the input and output data

open, close

open（［unit=］file equip.no , file=filename ［，control item］）

close（［unit=］file equip.no, file=filename ［，control item］）

CPU DISK

open(10,file=‘ data1’)


read(10) u1

write(20) u2

close(10)

close(20)

data1 u1

data2 u2



Input and output of data on the specified file

write（［unit=］ file equip.no ，［fmt=］format ［， control item ］）output list

read（［unit=］ file equip.no ，［fmt=］format ［， control item ］）output list

2. A large amount of data on hard disk

To connect or disconnect to hard disk


Identification of equipment

• Write the equip.no. first when ``unit” is not used

• File no. should be positive integer variable (include mathematical expression) or constant

• File name should be character variable or character constant

Control item

iostat= integer variable i normal i = 0

error i > 0

status=‘ string ‘ old : file exist

new : no other file with the same name

replace : overwrite on the same name file

unknown : new for input , replace for output

delete : delete after use （only for close statement）

if ``status=“ is skipped, d efault is unknown

position=‘string ‘ rewind : start from the beginning

append : add at the end of file

open, close

open（［unit=］file equip.no , file=filename ［，control item］）

close（［unit=］file equip.no, file=filename ［，control item］）


Identification of equipment

* console (key board)

Number file equipment no. specified by open statement

6 console (display)

Character variable inner file

Control item

iostat= integer variable i normal i = 0

error i > 0

end of file I < 0

advance=‘no’ no change of line after input or output

Input and output with file specification

write（［unit=］ file equip.no ，［fmt=］format ［， control item ］）output list

read（［unit=］ file equip.no ，［fmt=］format ［， control item ］）output list


character::d*10 Integer ::p=123456789 write(d,’(i10)’ ),p d=‘_123456789’ substituted character string at d

from the right

Inner file

Input and output without format

Very fast and efficient for input and output for binary data (large volume of data)

real*8, dimension(-10000:10000) :: a,b,c,d read(10) a,b write(20) c,d


Open (8,file=‘result’) Write(8,’(8f10.7)’) (x(i),i=1,8) close(8)

character(80) :: line(1:1000) open(11,file=‘../mydata/report.txt’) do j=1,1000 read(11,’(a)’,iostat=io) line(j) if(io<0) exit end do close(11)

character:: filename*10 print*, ‘Input file name?’ read ‘(a)’, filename open(10,file=filename)

Example 1

Leave do loop when the end is found


read(8,’(i10)’,iostat=i) k If(i>0) stop ‘Input file error’

character(4) :: fn . . . . irun=32 write(fn,'(i4.4)') irun open(10,file=‘int_'//fn//‘.dat') open(20,file=‘spect.dat’) . . . . write(10,’(1x, 2(f7.4,1x))’) time,esum . . . . write(20) (energy(i), i=1,kmax) close(10) close(20)

Example 2

Example 3

Inner file

Create new file ``Int_0032.dat”

Create new file ``spect.dat”

Write data at `` Int_0032.dat”

Write data at `` spect.dat”

iw.m : write w digit integer from the right when the number is shorter than m digit, 0 s are appended to the left. irun=32 0032


Miscellaneous

backspace（［unit=］file unit no., isostat=integer variable）

rewind（［unit=］ file unit no., isostat= integer variable ）

Back to the previous record point

Back to the head of the file

endfile（［unit=］ file unit no., isostat= integer variable ）

Write the end of file


!------ Character Code Table ------ integer ::i,j open(8,file='Code_table') do i=0,31 write(8,"(7(2x,i3,2x,a))")(i+32*j,char(i+32*j),j=1,7) end do close(8) end


７


Sub program

indispensable to make sturctured program

…..

call sub_A

call sub_B

…..

call sub_C

….

call sub_A

….

end

sub_B

sub_A

sub_C

call sub_G

Mail program Sub program Subroutine To do specified routine work only

can be used in other program

sub_G


subroutine example

Read three numbers and sort them in the ascending order

Main program

Sub routine

Dummy arguments

Actual arguments

implicit none integer :: i,j,k read *,i,j,k if(i<j) call swap(i,j) if(j<k) call swap(j,k) if(i<j) call swap(i,j) print *,i,j,k end ! ! ! subroutine swap(p,q) integer :: p,q,r r=p; p=q; q=r end


subroutine name ［(list of dummy arguments)］

……….. end subroutine ［name］

Subroutine

call subroutine_name［ (list of dummy arguments) ］

call

return

return

Set the return point to the main program

……… call sub(…,code) select case(code) case(1) …… case(2) …… default …… end select ……

subroutine sub(…,return_code) …… return_code=1 return …… return_code=2 return …… return_code=999 end subroutine sub

Variables, Arrays


Subprogram

・ Only one main program

・ No limits on the number of subprograms

・ Allowed to call other subprograms from one subprogram

・ Not allowed to call itself from subprogram

(except reccurence statement)

・ The number, types, size, shape of the actual and dummy arguments

must be identical


integer, parameter::n=100 real, dimension (0:n):: x,y ……… call set_initial(x,y,n) ……… End ! subroutine set_initial(a,b,m) integer :: m,i real, dimension(0:m)::a,b do i=0,m a(i)=0.0 b(i)=0.0 end do end set_initial

Number, type, shape of dummy arguments

Number, type, shape of actual arguments

call set_initial(y,x,n)

subroutine set_initial(a,b,m)

The number, types, size, shape of the actual and dummy arguments must be identical

Order of variables important

The data are passed according to the order

・Coinsident ot the numebr, types, shapes of arguments

・Order

・Correspondence between the variables in actual and dummy arguments

no need of the same name


Effective range of variables

Variable name, sentence no. and structure name are effective in one

program only

…..

call sub_A

….

y=sin(2.0*x)

….

print *, x,y

end

Subroutine sub_A

….

x=2.0*k

y=exp(-x)

…..

end subroutine sub_A

Main program Subprogram

No share of x,y

How share?


Method of sharing

１ aruguments

２ use common

３ use module

１．Arguements

call set_initial(x,y,z,n) subroutine set_initial(a,b,c,m)

2． common

common［/common block/］list

common/param1/x,y,z Common/param2/a,pi ….

common/param2/s,pi ….

Program 1 Program 3

common/param1/p,q,r ….

Program 2

(x,y,z) (p,q,r) (a,pi) (s,pi)

・number, types, shapes

・order

・no need to use the same name

(x,y,z,n) (a,b,c,m)

・number, types, shapes

・order

・no need to use the same name


save

No guarantee to preserve the value of variable defined

in subprogram for the next call

Use save statement

integer :: n real :: z do n=1,100 call sub_A(z,n) …… end do end

Main Program Sub program

subroutine sub_A(w,k) real, save :: pi …… if(k==1) then …… pi=3.14159265 end if w=cos(0.1*pi*k) …… end subroutine sub_A

Not sure if the value of pi is preserved unless save is made


Attribute of input or output of dummy arguments

Specify input (in) or output (ou) of arguments

type, intent（in, out, inout）：：list

intent

implicit none integer :: i,j,k read *,i,j,k if(i<j) call swap(i,j) if(j<k) call swap(j,k) if(i<j) call swap(i,j) print *,i,j,k end

subroutine swap(p,q) implicit none integer, intent(inout) ::p,q integer :: r r=p; p=q; q=r end


Example１

!------Drawing a Histogram -------- integer :: nn(0:9),j character(len=80) :: cn character(len=30) :: fmt nn=(/1,5,12,22,28,35,48,28,10,3/) fmt='(1x,i2,"-",i2,1x,a50)' do j=0,9 call str(nn(j),cn) print fmt,10*j,10*j+9,cn end do end ! subroutine str(n,c) integer, intent(in) :: n character(len=80), intent(out) :: c integer ::k c=‘ ‘ c(1:1)='I' do k=2,n+1 c(k:k)='#' end do end

Try to use comment ! c=‘ ‘

I#########

50-59 I############ write


Example ２

subroutine count(n) implicit none integer, intent(out) :: n integer :: i, sum=0 ! sum=0 do i=1,10 sum=sum+i end do n=sum end subroutine count ! Program main implicit none integer :: n call count(n) print '(1x,i6)',n call count(n) print '(1x,i6)',n end

Set initial value of sum

n=55

n=110

sum=55 in the previous step remains

Set sum=0 at each time


External function

Function defined by programer

External function： subroutine, external function

Built-in function ： Mathematical function already built-in the fortran

［type］ function name（argument list）［result（variable names）］

………

end function name

Function subprogram

function name（ argument list ）［result（ variable names ）］

type :: function name or result variable name

………

end function name

Function name (or result variable name)= formula

Substitution of function value


Function name (or result valuable name)= formula

Substitution of function value

integer :: n=100 real :: x,dx,sum=0.0,a=1.0 dx=1.0/n do i=1,n-1 x=i*dx sum=sum+fr(x,a) end do sum=(sum+0.5*(fr(0.0,a)+fr(1.0,a)))*dx print ‘(1x,e10.3)’,sum end ! real function fr(x,p) real :: x,z,p z=(p-x*x)/(p+x*x) fr=log(z) end function fr

Referring the external function

Function name （actual argument list）

Trapezoidal rule

・Coinsident ot the numebr, types, shapes of arguments

・Order

・Correspondence between the variables in actual and dummy arguments

no need of the same name


!-- Generating a Hexadecimal Representation ------- function hex(n) character(len=8) :: hex character(len=1) :: h(0:15)=(/ '0','1','2','3','4','5','6','7',& '8','9','A','B','C','D','E','F'/) integer, intent(in) ::n integer :: nin,j,nn hex=' ' nin=n do j=8,1,-1 nn=nin/16 hex(j:j)=h(nin-16*nn) if(nn==0) exit nin=nn end do end function hex !---- Main ---- program main implicit none character(len=8) :: hex integer :: i print *, 'Input a positive Integer or negative one to stop:' do read *, i; if(i<0) exit print *,hex(i) end do end

From the right

Example

Input n

Hexadecimal expression


type, external :: function list

external :: function list

Declaration of function attributes

Let the compiler know that the function is external

external

type, intrinsic :: function list

intrinsic :: function list

intrinsic The external function is chose n when the function name is the

same as the built-in function

call integral(gamma,0.0,1.0,s) ………… real function gamma(x) …………

Declaration of attribute is neccessary for function argument


implicit none real:: h, phix,err real, external ::Gaussian print *, 'What is your deviation value?' read *, h h=(h-50.0e0)/10.e0 call simpson(Gaussian, 0.0e0,h, 16,phix) err=(0.5e0-phix)*100.0e0 print '(1x,a,f6.2,a)', & 'From the top you are ', err, ' %.' end subroutine simpson(func,a,b,kubun,integ) implicit none integer, intent(in)::kubun integer:: i real, external :: func real, intent(in) :: a,b real, intent(out) :: integ real :: x,dx dx=(b-a)/(2.0e0*kubun) integ=func(a)+4.0e0**func(a+dx)+func(b) do i=1,kubun-1 x=a+(2.0e0*i)*dx integ=integ+2.0e0*func(x)+4.0e0*func(x+dx) end do integ=integ*dx/3.0e0 end subroutine simpson

real function Gaussian(s) implicit none real, parameter :: rn=2.506628e0 ! \sqrt(2*pi) real, intent (in) :: s Gaussian=exp(-s*s/2.0e0)/rn end function Gaussian

Simpson‘s formula

x

Example


Contains

To include subroutines and functions in one program unit

integer :: n real :: z,g,pi pi=3.14159265 do n=1,100 call sub_A(z,n) …… g=func(z) …… end do ……… Contains subroutine sub_A(x,m) ……… z=log(func(x)) ……… end subroutine sub_A ! real function func(x) ……… func=0.5*exp(-pi***0.25d0) end function func end

The same rules for subroutine and function apply

contains

１．No declaration of type and attributes of internal function

２．Variables and arrays defined in the parent program are

effective in the child program and the same values are shared

３．When the same variable names as in the parent program are declared,

they are effective only in the child program and not shared.

４．Conatins statement is not allowed in the internal procedure

５．Referring other internal procedure from one internal procedure

is allowed when the two are in the same parent program

６. There must be ``subroutine, function“ in the end statement in

the contains program


Example

!----- All Round Calendar ------ program main character(len=9) :: name(0:6)=(/'Sunday ', & 'Monday ', 'Tuesday ','Wednesday', & 'Thursday ','Friday ','Saturday '/) integer :: md(1:12)=(/31,28,31,30,31,30,31,31,& 30,31,30,31/) integer :: y,m,d,ld do print *,'year?' read *,y; if(y<=0) exit md(2)=28+leap(y) print *,'month?‘ do read *,m; if(0<m .and. m<=12) exit print *, 'm<=12' end do print *, 'Day?' read *, d if(d<=0 .or. d>md(m)) then print *, 'nothing' else call count(ld) print "(1x,i5,',',i2,',',i2,',',a9,a/)", & y,m,d,name(ld) end if end do

! contains ! Count how many days since Jan. 1, 0001 ! and compute what day is subroutine count(n) integer, intent(out) :: n n=yday(y)+sum(md(1:m-1))+d n=mod(n,7) end subroutine ! ! Count how many dasys on Dec. 31 of ! the last year since Jan 1, 0001 function yday(x) result(days) integer :: x,days days=365*(x-1)+(x-1)/4-(x-1)/100+(x-1)/400 end function yday ! integer function leap(x) ! 1 for leap year ! 0 otherwise integer :: x leap=yday(x+1)-yday(x)-365 end function leap ! end program


Sharing of arrays

quite often used in numerical computation

pass the arrays in the main program to subprograms accoring to

their shape and size

１．explicit shape array

２．assumed shape array

array of explicit shape with fixed upper and bottom boundary (static)

array that has no size specification except shape in the declaration

statement of subprogram and its size is fixed by the size of the actual

arguments when it is referred. (dynamic)


Explicit shape array

Array of explicit shape with fixed upper and bottom boundary (static)

Subroutine sub(x,n)

integer, intent(in):: n

integer :: a(1:n),j

real, intent(out) :: x

real, allocatable ::temp(:)

allocate(temp(0:n))

１．Dummy integer variables which are passed from the parent program

２．Value obtained by the size function applied to dummy array

３．Integer variable shared by common statement

４．Integer variables of the internal procedure that are shared with those of

the parente program

Method of passing size

a(1:n) and temp(:)

are not the dummy arrays but

used only in the subroutine


program main integer, allocatable :: number(:) integer ::n,m,i print *,‘How many applicants？’ read*,n No. Of applicants print *, ‘How many winners?’ read*,m No. Of winners allocate(number(n)) call chusen(number,n,m) ! print *,‘順位当選番号’ do i=1,min(n,m) print '(2x,i4,i8)',i,number(i) end do end subroutine chusen(kk,n1,n2) integer, intent(in) :: n1,n2 integer, intent(out) :: kk(n1) integer :: i,ir,j,temp real ::x kk=(/(i,i=1,n1)/) print *,'ir' read *, ir do i=1,min(n1,n2)-1 call ran(ir,x) j=i+int(x*(n1-i+1)) temp=kk(i) kk(i)=kk(j) kk(j)=temp end do end subroutine

Example

subroutine ran(i,r) integer, intent(inout):: i real, intent(out) :: r integer, parameter ::mask=2147483647,a=48828125 i=iand(a*i,mask) r=real(i)/real(mask) end subroutine

Generate pseudo

random number (0,1)

Seed of random number (enter any positive integer

Choose n winners out of m applicants

Random number （0,1）

Choose one out of the residuals

Swap I and j


Assumed shape array

array that has no size specification except shape in the declaration

statement of subprogram and its size is fixed by the size of the actual

arguments when it is referred. (dynamic)

integer, parameter:: n=100 real :: u(0:n,0:n),x(0:n),y(0:n) interface subroutine fmake(w,a,b) real, intent(in)::w(0:,0:),a(0:) real, intent(out)::b(0:) end subroutine fmake end interface ………… call fmake(u,x,y) ………… end

Parent program Sub program

subroutine fmake(w,a,b) real, intent(in)::w(0:,0:),a(0:) real, intent(out)::b(0:) integer ::m Real :: sum m=ubound(w,1) do i=0,m sum=0.0 do j=0,m sum=sum+w(i,j)*a(j) end do b(i)=sum end do end subroutine fmake

Upper limit in the first

dimension of w

Array size is determined by the

size of the actual argumenst

Interface statement

Let the compiler know the program structure in advance


Subroutine test(x) implicit none real :: x(:) ! real :: x(0:) print ‘(2i3)’, lbound(x), ubound(x) end subroutine test Program main implicit none integer :: I real :: s(-10:10)=(/(i,i=1,21)/) interface subroutine test(x) real :: x(:) end subroutine test end interface call test(s) call test(s(0:8)) End

Return 1 and 21 Return 1 and 9

Example 1

Interface

procedure name to be referred

shape, attributes of dummy arguments

end interface

interface structure


program main integer :: n,i real, allocatable :: a(:,:) interface function trace(x) result(tr) real, intent(in) :: x(1:,1:) real :: tr end function end interface print *,'n?' read *,n allocate(a(1:n,1:n)) do i=1,n ! print '(1x,a,i2)','line',i do j=1,n print '(1x,a,i2,a,i2,a)', & 'Enter element a(',i,' ,',j,')' read *,a(i,j) end do end do print '(1x,a,F7.4)','trace=',trace(a) end

Example Trace of square matrix

!----Trace of a Square Matrix function trace(x) result(tr) Implicit none integer :: n1,n2,i real, intent(in) :: x(1:,1:) real :: tr n1=size(x,1) ! See Text p.82 n2=size(x,2) if(n1/=n2) then print *, 'not square';return else tr=0.0 do i=1,n1 tr=tr+x(i,i) end do end if end function


Arrays in contains

Allowed to use the same integer variable names as those defined in parent program

implicit none real, dimension(1:3) :: a,b,c,bc print *, "Input three vector's components" read *, a read *, b read *, c bc = vector_prod(b,c) print '( "Sp3 = ",F10.3)', scalar_prod(a,bc) print '( "Vp3 = (",3F10.3,")")', vector_prod(a,bc) contains function vector_prod(x,y) result(z) implicit none real, dimension(1:3), intent(in) :: x,y real, dimension(1:3) :: z z(1)=x(2)*y(3)-x(3)*y(2) z(2)=x(3)*y(1)-x(1)*y(3) z(3)=x(1)*y(2)-x(2)*y(1) end function vector_prod function scalar_prod(x,y) result(sp) implicit none real, dimension(1:3), intent(in) :: x,y real :: sp sp=x(1)*y(1)+x(2)*y(2)+x(3)*y(3) end function scalar_prod end

No ``intent”

Example: Triple products of scalar and vectors

Review of Contains

real, external :: vector_prod, scalar_prod はいらない

b x c a . (b x c) a x (b x c)

１．No declaration of type and attributes of internal function

２．Variables and arrays defined in the parent program are

effective in the child program and the same values are shared

３．When the same variable names as in the parent program are declared,

they are effective only in the child program and not shared.

４．Conatins statement is not allowed in the internal procedure

５．Referring other internal procedure from one internal procedure

is allowed when the two are in the same parent program

６. There must be ``subroutine, function“ in the end statement in

the contains program


Recursive call

Subprogram calls itself (do not fall in the infinite loop)

recursive subroutine subroutine name（list of dummy arguments）

recursive function name （list of dummy arguments） result（variable names）

Specifying recursive procedure

must!


Legendre polynomials

Chebyshev polynomials

Hermite polynomials

n=0

n=1

n=2

n=3

n=4

n=0

n=1

n=2

n=3

n=4

n=0

n=1 n=2

n=3

n=4


program main implicit none integer :: n,i real :: x,y,p character(len=1) :: c(-25:25) print *,'What is the order of Polynomial, n?' read *,n print '( "Legendre Polynomial P(",i2,",x)")',n do i=-20,20 x=0.05*i ! dx=1/n=0.05 y=p(n,x) call presentation print '(f5.2,2x,f10.6,1x,51a1)',x,y,c end do contains subroutine presentation integer ::j if(i/=0) then c=' ' else c=(/('-',j=-25,25)/) end if c(-25)='I';c(0)='I';c(25)='I' c(nint(25*y))='*' end subroutine end

recursive function p(n,x) result(pnx) real :: pnx integer, intent(in) ::n real,intent(in) ::x if(n==0) then pnx=1.0 else if(n==1) then pnx=x else pnx=(2.0-1.0/n)*x*p(n-1,x) & -(1.0-1.0/n)*p(n-2,x) end if end function

Legendre polynomials

Quite often appears in the solution of partial diff. Eq.

Recurrence relation


! Computation of special functions integer, parameter:: fun_kind=3 ! Choose type of special function ! fun_kind=1: Legendre, =2: Chebyshev, =3: Hermite real*8, parameter:: x_max=1.d0 Integer, parameter:: n_grid=50, n_order=4, k_write=2 character(len=1):: cn(0:9)=(/'0','1','2','3','4','5','6','7','8','9'/) character(len=2):: fn(1:3)=(/'p_','t_','h_'/) real*8 :: dx,x,y,fun integer :: i,n,n_n,n_p,n_extend select case (fun_kind) case(1) ; print *,' Legendre Polynomial P_n(x) ' case(2) ; print *,' Chebyshev Polynomial T_n(x) ' case(3) ; print *,' Hermite Polynomial H_n(x) ' end select dx=x_max/n_grid n_extend=1 if(fun_kind==3) n_extend=4 ! extend the x-domain for Hermite polynomial n_n=-n_grid*n_extend n_p= n_grid*n_extend do n=0,n_order write(6,'(/," n=",i2,/)') n open(10+n,file=fn(fun_kind)//cn(n)//'.dat') do i=n_n,n_p x=i*dx y=fun(fun_kind,n,x) write(10+n,'(1pe10.3, 2x, 1pe10.3)') x, y if(mod(i,k_write)==0) write(6,'(1pe10.3, 2x, 1pe10.3)') x,y end do end do end

Legendre, Chebyshev, Hermite polynomials

1/2


recursive real*8 function fun(kind,n,x) result (sf) integer, intent(in) :: kind, n real*8, intent(in) :: x if(n==0) then sf=1.d0 else if(n==1) then select case(kind) case(1); sf=x case(2); sf=x case(3); sf=2.d0*x end select else select case(kind) case(1); sf=(2.d0-1.d0/n)*x*fun(kind,n-1,x)-(1-1.d0/n)*fun(kind,n-2,x) case(2); sf=2.d0*x*fun(kind,n-1,x)-fun(kind,n-2,x) case(3); sf=2.d0*x*fun(kind,n-1,x)-2.d0*(n-1)*fun(kind,n-2,x) end select end if end

2/2


Report ４

1. To numerically integrate the following functions by using subroutine, external

function and recurrence procedure

2. To compare the numerical results with the analytical calculations

Report No.

ID no.

Name

Date Use Simpson’s formula

（１）

（２）

Use trapezoidal forumla

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